Problem 466

Let $N(X,\delta)$ denote the maximum number of points $P_1,\ldots,P_n$ which can be chosen in a circle of radius $X$ such that

\| \lvert P_i-P_j\rvert \| \geq \delta

for all $1\leq i<j\leq n$ . (Here $\|x\|$ is the distance from $x$ to the nearest integer.)

Is there some $\delta>0$ such that

\lim_{x\to \infty}N(X,\delta)=\infty?

Second statement

Let $N(X,\delta)$ denote the maximum number of points $P_1,\ldots,P_n$ which can be chosen in a circle of radius $X$ such that

\| \lvert P_i-P_j\rvert \| \geq \delta

for all $1\leq i<j\leq n$ . (Here $\|x\|$ is the distance from $x$ to the nearest integer.)

Is there some $\delta>0$ such that

\lim_{X\to \infty}N(X,\delta)=\infty?

Source: erdosproblems.com/466

Number theory

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

PROVED, the site's label (page last edited 16 September 2025), which describes the corrected Statement: the commentary credits Graham's construction and Sárközy's power lower bound. The result is recorded as an accepted full claim: Sárközy's Theorem 1 of [Sa76] Part II (Studia Sci. Math. Hungar. 11 (1976), refereed; p. 106) gives $N(X,\delta)>X^{1/2-\delta^{1/7}}$ for $0<\delta\le1/(6\cdot8^4)$ and $X$ large depending on $\delta$ , so $N(X,\delta)\to\infty$ as $X\to\infty$ for every such $\delta$ and in particular for $\delta=1/(6\cdot8^4)$ ; the same paper (pp. 105--106) reports Graham's construction, the points $(x_i,x_i^2)$ with $x_1=10$ , $x_n=2x_{n-1}$ , which gives $N(X,1/10)>\frac1{10}\log X$ for large $X$ , the result the site's commentary credits to Graham. Graham's argument is reported by Sárközy with a sketch, while Erdős (1980, 1982) reports Graham's bound without a construction, and no separate paper of Graham's was located, so the claim page rests first-hand on Sárközy's Theorem 1 and reports Graham's construction second-hand, as Sárközy gives it, without depending on it. The derived standing is solved, proved.

Notes

The site's display takes its limit in a variable $x$ that does not occur in $N(X,\delta)$ , so read literally the display does not take the limit in the radius, and a comment in the site's discussion thread (11 May 2026) asks for the capital letter. The change replaces the lowercase $x$ under the limit by the radius $X$ , the only variable the expression contains; nothing else changes. The evidence is Erdős's own statement of the question in his 1982 survey [Er82e], Chapter II, §9, printed p. 67, which writes the radius as $x$ throughout and conjectures " $N(x,\delta)\to\infty$ ", the limit in the radius; Sárközy, who states the question as Erdős's conjecture, writes " $\lim_{X\to+\infty}N(X,\delta_0)=+\infty$ " in Part II of [Sa76], printed p. 105, and as (4) in Part I, printed p. 37. The defect is not the site's alone: the 1980 monograph [ErGr80], printed p. 93, prints " $\lim_{x\to\infty}N(X,\delta_0)=\infty$ " with the lowercase $x$ the site reproduces, so the misprint is the monograph's and the site's wording copies it. The site's commentary, which credits Graham and Sárközy with proving the question, answers the corrected statement. The standing judges the corrected Statement.

Sa76: Sárközy, A., On distances near integers. I, II. Studia Sci. Math. Hungar. 11 (1976), 37--50 (received 10 December 1975) and 105--111 (received 11 February 1976). Part II: Graham's construction, pp. 105--106; Theorem 1, p. 106; Corollary, p. 107; the remark $N(X,1/10)>X^C$ , p. 110. Part I: conjecture (4) and Graham's proof of it attested, p. 37. Library homes: Sárközy 1976 and Sárközy 1976 (both in the REAL-J open scan of the whole 1976 volume).
Er72: Erdős, P., Extremal problems in number theory. Proceedings of the Number Theory Conference (Univ. Colorado, Boulder, 1972), 80--86. Section IV, printed p. 83. Library home: Erdős 1972.
ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980). Printed pp. 92--93. Library home: Erdős and Graham 1980.
Er82e: Erdős, P., Some of my favourite problems which recently have been solved. Proceedings of the International Mathematical Conference (Singapore, 1981), North-Holland Math. Stud. 74 (1982), 59--79. Chapter II, §9, printed pp. 67--68. Library home: Erdős 1982.
Ko01: Konyagin, S. V., On the distances between points on the plane. Mat. Zametki 69 (2001), no. 4, 630--633 (in Russian; Math. Notes 69 (2001), 578--581). Its introduction (p. 630) attests Graham's proof and Sárközy's lower bounds; its Theorem is the upper bound of Problem 465. Library home: Konyagin 2001.
GoMo26: Goenka, R. and Moore, K., Point sets avoiding near-integer distances. arXiv:2605.06621v1 (7 May 2026). A lead on the higher-dimensional analog; only its abstract was read.

Problem 466

Second statement

Status

Notes

Proof claims

Discussion